Characterisations of Ideal Threshold Schemes

Since 1979 when threshold schemes were introduced by Blakley (Bla79) and Shamir (Sha79), many papers in this area have been published. These papers are mostly about particular designs and applications of threshold schemes. In this work, we are interested in characterisations of ideal threshold schemes rather than particular constructions. Of course, these characteristic properties do not depend on any form of any ideal threshold scheme. This enables us to look at any ideal threshold scheme from new points of view, and further find new results on ideal threshold schemes. This work is structured as follows. The basic concepts of threshold schemes, and more generally, secret sharing schemes are introduced in Section 2. In Section 3, we describe (weakly and strongly) perfect secret sharing schemes using defining matrices. By the original definition (BS92), an ideal threshold scheme is a strongly perfect threshold scheme for which the set of secrets and the set of shares have the same cardinality. In Sections 4 and 5, we derive a series of properties of weakly perfect threshold schemes that are helpful for us to characterise ideal threshold schemes in Section 6 from different points of view. Consequently “strongly perfect” in the original definition of ideal threshold schemes can be replaced by “weakly perfect”. We further obtain more results on ideal threshold schemes in Section 7 including an application in cheating prevention.